Hilbert schemes for moduli spaces
The Hilbert scheme of P^n classifies nice families of subschemes. It is a basic object in an algebraic geometer's toolbox. Consequently, for a general scheme one might be interested in whether or not a Hilbert scheme can be constructed. If the scheme in question is separated, it is a classical result that there is a Hilbert scheme. When one studies moduli spaces, however, you cannot always assume that your moduli space is separated. I will cover some ideas that allow you to obtain a Hilbert scheme for non-separated moduli spaces.